Hebrew calendar arithmetic requires that the time calculated for the
molad does not exceed the first day of any month. In the calendar's present
format, the molad zaqen postponement rule is used to guard against
any such excess.

The correction made by the molad zaqen rule applies only to
Rosh Chodesh of Kislev or Shevat of the previous year.
Consequently, this rule is in no way related to the visibility of the new
moon on Rosh Hashannah.

The excess of time of the molad over the first day of any month will be
referred to as the overpost.

The overpost problem arises when the leap month of 30 days is placed anywhere
after the month of Heshvan during a deficient or regular leap year (ie, years
of either 383 or 384 days).

Because it was decided at some point in the calendar's history to place the
leap month of 30 days as the sixth month of any leap year, corrective
action also had to be made to overcome the overpost.

Calendar arithmetic shows that postponing Rosh Hashannah to the next
allowable day whenever the molad of Tishrei exceeds 18h;656p on one of the
permissible days for the holiday eliminates the overpost for the immediately
preceding year.

The molad zaqen rule sets that postponement time at or after exactly
18h (noon) of the day.

Demonstrating the Overpost

For some Hebrew year H

Let R = the number of days that have elapsed up to the first of Tishrei H
since day 0 of year 1H
Let t = the time of the molad on day R
Let R' = the number of days that have elapsed up to the first of Tishrei H+1
since day 0 of year 1H
Let t' = the time of the molad on day R'
Let A = its annual lunation period
Let L = its length in days
Let d' = 1d - 1p = 23h 1079p

Let u = the molad period = 29d 12h 793p
Let i = the number of months that have elapsed in year H
Let m(i) = the ith month in year H
Let s(i) = the number of days that have elapsed in year H up to the start
of month m(i+1)
Let D(i) = i*u - s(i) for any month m(i+1)
Let E(i) = the excess time of the molad on the first day of any
month m(i+1) in year H

Then R + t + i*u = time of the molad for month m(i+1)R + s(i)= number of days that have elapsed up to the start
of month m(i+1)

E(i), the excess time of the molad on the first day of any month in year H,
is the difference between those two values.

Hence, E(i) = t + i*u - s(i) = t + D(i)

The following table shows the value of D(i) for any given month m(i+1) in
any year of length L.

The table follows the traditional placement of the leap month of 30 days
as the sixth month of the leap year.

A minus sign following the time of D(i) indicates a negative value for D(i).

VALUES FOR D(i)

YEAR LENGTH L IN DAYS

MONTH

353 days

354 days

355 days

383 days

384 days

385 days

Heshvan

11h 287p-

11h 287p-

11h 287p-

11h 287p-

11h 287p-

11h 287p-

Kislev

1h 506p

1h 506p

22h 574p-

1h 506p

1h 506p

22h 574p-

Tevet

14h 219p

9h 861p-

1d 9h 861p-

14h 219p

9h 861p-

1d 9h 861p-

Shevat

1d 2h 1012p

2h 1012p

21h 68p-

1d 2h 1012p

2h 1012p

21h 68p-

Adar

15h 725p

8h 355p-

1d 8h 355p-

15h 725p

8h 355p-

1d 8h 355p-

v'Adar

4h 438p

19h 642p-

1d 19h 642p-

Nisan

1d 4h 438p

4h 438p

19h 642p-

17h 151p

6h 929p-

1d 6h 929p-

Iyar

17h 151p

6h 929p-

1d 6h 929p-

5h 944p

18h 136p-

1d 18h 136p-

Sivan

1d 5h 944p

5h 944p

18h 136p-

18h 657p

5h 423p-

1d 5h 423p-

Tammuz

18h 657p

5h 423p-

1d 5h 423p-

7h 370p

16h 710p-

1d 16h 710p-

Av

1d 7h 370p

7h 370p

16h 710p-

20h 83p

3h 997p-

1d 3h 997p-

Elul

20h 83p

3h 997p-

1d 3h 997p-

8h 876p

15h 204p-

1d 15h 204p-

Determining the Overpost

Using the following two conditions, values of D(i) which do not
lead to overposts can be found.
1. When D(i) < 0 there can be no overpost for any value of t.
When D(i) < 0 , t + D(i) < d' since t <= d' (by def'n).
2. When d' + A - D(i) > L + d' there can be no overpost for any value of t.
By definition t + D(i) = d' + O(i)
so that t - O(i) = d' - D(i)t - O(i) + A = d' - D(i) + A
When d' + A - D(i) > L + d' then
t + A - O(i) > L + d' since t - O(i) = d' - D(i)
hence, - O(i) > L + d' - (t + A)
now L + d' => t + AL + d' - (t + A) => 0- O(i) > L + d' - (t + A) => 0 O(i) < 0 (the overpost value O(i) is negative!)
Therefore, when d' + A - D(i) > L + d' there can be no overpost since
the overpost value O(i) is less than zero.
From the above, it is necessary only to look for positive values of D(i)
for which d' + A - D(i) < L + d'
To avoid a potential overpost, it is necessary that the maximum value of t
be decreased by O(i). That leads to the following relationships
R + (t - O(i)) + A = R + (d' - D(i)) + A = R' + t'd' - D(i) + A + (R - R') = d' - D(i) + (A - L) = t'
If more than one value for t' is found, then the smallest value of
t' would become the limiting value for the maximum permissible time of the
molad on Rosh Hashannah. Otherwise an overpost will occur.
The table for the values of D(i) may now be scanned.

For L = 353 Days

The largest excess for the 353 day year is 1d;7h;370p. Applying the formula
in condition 2,d' - D(i) + A cannot fall below 354d; 1h;505p.
Since this value is larger than 354d - 1p, the overpost problem cannot arise
for the largest possible excess time value. Hence, the overpost cannot take
place in a 353 day year.

For L = 354 Days

The largest excess for the 354 day year is 7h;370p. Applying the formula in
condition 2,d' - D(i) + A cannot fall below 355d; 1h;505p. Since this
value is larger than 355d - 1p, the overpost problem cannot arise for
the largest possible excess time value. Hence, the overpost cannot take place
in a 354 day year.

For L = 355 Days

All of the excesses in the 355 day year are negative. So according to
condition 1, there is no possibility of an overpost problem in a 355 day
year.

For L = 383 Days

Applying the largest excess value of 1d;2h;1012p to the formula in
condition 2, the result is 383d;18h;656p. Consequently, corrective action
must be taken to prevent an overpost from occurring after the first day of
Shevat in such a year.

Hence, 18h;656p becomes one of the values of t' to be considered for
the maximum allowable time of the molad on Rosh Hashannah.

In the 383 day year, the second largest molad excess is 20h;83p. Applying the
formula ofcondition 2, the result of d' - D(i) + A cannot fall below
384d;1h;505p. Since this value is larger than 384d - 1p, the overpost problem
cannot arise for any other month in the 383 day year.

For L = 384 Days

In the 384 day year only two values are positive, and both of these values
when used in the formula of condition 2 yield results which indicate a molad
overpost problem taking place in both the months of Kislev and Shevat. In
this case, the smaller of the two times possible for t' is 18h;656p.

For L = 385 Days

All of the molad excesses in the 385 day year are negative and so cannot
lead to a molad overpost.

Final Value of t'

The above shows that 18h;656p must be the maximum allowable time for the
molad onRosh Hashannah of the subsequent year if an overpost is to be avoided.

The scholars, possibly to be on the safe side, chose an even lesser time of
18h for the molad zaqen postponement rule. However, it may also have been
chosen for the reason that with the 18h limit exactly 1 out every seven
years will be postponed due to the molad zaqen rule, thus bringing together
two numbers of considerable significance in Jewish numerological traditions.

Examples Correcting for the Overpost

The time for molad of Tishrei 76H is 27,377d;1h;915p
and corresponds to Saturday 21 August -3685g. The time of this molad
bypasses all of the postponement rules.

The year 76H is a leap year. So the next molad of Tishrei (77H) is

27,377d;1h;915p + 383d;21h;589p = 27,760d;23h;424p ==> 5d;23h;424p

If the molad zaqen rule is not applied to the molad of Tishrei 77H then
the year 76H is 2 days shorter and must be 383 days long. On that basis
it may be seen that Rosh Chodesh Shevat will occur on day 27,494d and end at
27,494d;23h;1079p. Now, the molad of Shevat will be at

Subtracting the maximum time of Rosh Chodesh Shevat from the time of the
molad of Shevat, the result is

27,495d;4h;847p - 27,494d;23h;1079p = 4h;848p

From that simple calculation, the molad of Shevat of 76H can be seen to
occur 4h;848p after the end of Rosh Chodesh Shevat.

Because of its molad timing of Saturday;1h;915p, Rosh Hashannah 76H cannot
be postponed. Therefore the correction for the overpost must come from
a postponement of the year 77h. The molad zaqen rule can be applied to the
year 77H because its molad of Tishrei is on Thursday past 18h. That causes
two extra days to be inserted into the year 76H, and so, no Rosh Chodesh for
76H will end prior to the time of its corresponding molad.

A similar set of arithmetic can be found to take place for the molad
of Tishrei 5874H, corresponding to Monday 11 September 2113g. The molad
of Tishrei will arrive at 2d;1h;51p.

The molad of Shevat will occur 3h;1068p past Rosh Chodesh unless some
corrective action is taken. Since there can be no postponement for
Rosh Hashannah 5874H, the corrective action must come from a postponement of
Rosh Hashannah 5875H.

The molad of Tishrei 5875H is 0d;22h;640p. Since the time is on Saturday past
18h, the molad zaqen rule can be applied causing 5875H to be postponed
by 2 days. These two days will be added to the months of Heshvan and Kislev,
in 5874H, thereby correcting the overpost found for the month of Shevat.

From these examples it is clear that the molad zaqen rule is an arithmetical
device that applies a necessary correction only to the previous year.
Therefore the rule has nothing whatever to do with the visibility of the moon
on Rosh Hashannah.

Eliminating The Molad Zaqen Rule

The molad zaqen rule is the corrective action needed to overcome the
overpost problem when the leap month is placed as the sixth
month of a leap year.

A simple scan of the 353 and 354 day years in the above table shows that if
the leap month is placed prior to the month of Heshvan, then
absolutely NO corrective action is required. Hence, the molad
zaqen rule can be eliminated.

Similarly, a simple scan of the 353 and 354 day years shows that if the
leap month is placed past the month of Adar, then the corrective action will
have to be the postponement of the subsequent Rosh Hashannah for times that
are earlier than currently specified in the molad zaqen rule.

The need for the molad zaqen rule could also have been eliminated if the
month of Heshvan had been made a 30 day month, and a day removed from some
month past Adar I, such as the mournful month of Av, to be returned to the
month in abundant years.

Maintaining The Keviyyot

To maintain the existing keviyyot when the molad zaqen is eliminated,
it is necessary to add 6 hours to the time specified in both Dehiyyah GaTaRad
and Dehiyyah BeTU'TeKaPoT.

Hence, the limiting times of these two rules become 15h;204p and 21h;589p
respectively.

Once the adjustments are made, it is possible to develop the following
distribution table of the keviyyot over the full calendar cycle of
689,472 years.

Keviyyot - Without the Molad Zaqen Rule

YEAR LENGTH IN DAYS

DAY

353

354

355

383

384

385

TOTALS

Mon

39369

0

81335

40000

0

32576

193280

Tue

0

43081

0

0

36288

0

79369

Thu

0

124416

22839

26677

0

45899

219831

Sat

29853

0

94563

40000

0

32576

196992

TOTALS

69222

167497

198737

106677

36288

111051

689472

The only change to the calculated results is that the number of Rosh Hashanot
that are not postponed is increased by 98,496 from 268,937 to 367,433. The
increase comes from the loss of the 98,496 molad zaqen postponements over
the full calendar cycle of 689,472 years.

The Molad Overpost Distribution

The above distribution table does not show that in some of the years
the time of the molad can exceed either the first day of the month of
Kislev, or the first day of the month of Shevat by as much
as 5 hours and 422 parts. And, in some years the overpost will take place
in both of these two months.

The overpost occurs in no other month.

Over the full calendar cycle the overpost is found to occur in 16,304
deficient leap years (383 day years) and in 4,440 regular leap years
(384 day years).

Over this same cycle, the overpost will occur in both the months of Kislev
and Shevat 2,220 times, and only 2,220 times for the month of Kislev.

Some Overpost Years

Assuming calendar calculations with all the rules in place except for the
molad zaqen, then for the range of years from 3760H (0g) to 4180H (420g),
the overpost can be seen to occur in the following years:-

OVERPOST FOR THE SPAN 0g to 420g

YEAR

OVERPOST

MONTH

YEARLENGTH

3773H

13g

3h 565p

Shevat

383 days

3844H

84g

219p

Shevat

383 days

3855H

95g

4h 67p

Shevat

383 days

3860H

99g

127p

Kislev

384 days

3860H

100g

1h 633p

Shevat

384 days

3922H

162g

824p

Shevat

383 days

3933H

173g

4h 672p

Shevat

383 days

4004H

244g

1h 326p

Shevat

383 days

4020H

260g

2h 740p

Shevat

383 days

4102H

342g

3h 242p

Shevat

383 days

4107H

347g

808p

Shevat

384 days

4118H

358g

4h 656p

Shevat

383 days

4180H

420g

3h 847p

Shevat

383 days

The range was selected because it covers a period during which the Talmud
was being developed.